Question

Find the area of the surface. the part of the plane 13x + 5y + z = 65 that lies in the first octant

Answer 1

The area of the surface in the first octant bounded by the plane equation 13x + 5y + z = 65 is 65 square units.

Step-by-step explanation:

To find the area of the surface in the first octant bounded by the plane equation 13x + 5y + z = 65, we need to determine the limits within the first octant.

In this context, the first octant refers to the positive values of x, y, and z.The equation of the plane represents a region in three-dimensional space, and within the first octant, x, y, and z are positive.

Setting up the integral bounds for the region, we consider the constraints placed by the equation in terms of x, y, and z. By determining the intercepts of the plane with the axes (where x, y, and z are 0), we establish the boundaries for integration.

Solving for these intercepts, we derive the limits for x, y, and z within the first octant. Using these limits, we set up the integral for the surface area, considering the differential elements in the x-y, y-z, and x-z planes.

Integrating over these limits yields the total area of the surface within the first octant bounded by the given plane equation. Evaluating the integral gives us the final area of 65 square units.

Answer 2

To find the area of the surface in the first octant defined by the plane 13x + 5y + z = 65, one must calculate the area of a triangular region identified by intercepts on the coordinate axes, resulting in an area of 32.5 square units.

Step-by-step explanation:

The question asks to find the area of a surface within the first octant of the plane defined by the equation 13x + 5y + z = 65. To solve this, we must determine the intercepts on the x, y, and z axes which occur when the other two variables are zero.

These intercept points are (5, 0, 0), (0, 13, 0), and (0, 0, 65) for the x-, y-, and z-axes respectively. We then construct a triangular region in the first octant bounded by these intercepts and the coordinate planes.

The area of the triangle can be found using the formula for the area of a triangle A = 0.5 * base * height. Here, the base would be the distance on the x-axis and the height would be the distance on the y-axis. So, we calculate the area as 0.5 * 5 * 13 which equals 32.5 square units.